Research notes on every problem, and a library of the papers behind them. Built from the open erdos repository.

Updated Oct 8

Problem 438

claims/: The 1 claim page of Problem 438, one per claimant's result; the problem's standing derives from them.

Statement. How large can $A\subseteq \{1,\ldots,N\}$ be if $A+A$ contains no square numbers?

Status. Solved: the largest such $A$ has $(11/32+o(1))N$ elements. The upper bound is Theorem 3 of Khalfalah, Lodha and Szemerédi (DIMACS report of December 2000; Discrete Math. 256 (2002), a refereed journal), and the lower bound is Massias's construction of density $11/32$ , after Lagarias, Odlyzko and Shearer had proved $11/32$ sharp for unions of residue classes and the general bound $0.475N$ . The site labels the problem SOLVED and credits the paper (page last edited 7 April 2026). Claim page: Khalfalah, Lodha and Szemerédi 2000 (accepted).

Source. erdosproblems.com/438, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #438, https://www.erdosproblems.com/438.

References.

[KLS02] [[../library/integer_sequences/khalfalah_2002_tight_bound_density_sum_no_two_perfect_square/_index|Khalfalah, A. and Lodha, S. and Szemerédi, E., Tight bound for the density of sequence of integers the sum of no two of which is a perfect square]]. Discrete Math. (2002), 243-255.
[LOS83] Lagarias, J. C. and Odlyzko, A. M. and Shearer, J. B., On the density of sequences of integers the sum of no two of which is a square. II. General sequences. J. Combin. Theory Ser. A 34 (1983), no. 2, 123--139, doi:10.1016/0097-3165(83)90051-1.

Formalization. Statement in formal-conjectures, ErdosProblems/438.lean (added 19 September 2026), with a sorry body, marked research solved; its formal_proof attribute points at Erdos438.lean in Boris Alexeev's lean-proofs repository, a Lean development that declares itself a formalization of a solution to the problem, names Khalfalah, Lodha and Szemerédi as its informal authors and Codex and GPT-5.6 Sol as its formal authors, and proves that the extremal density tends to $11/32$ . The community database records the statement as formalized since the same date. Neither file was built or audited here; the claim page links the Lean development at a pinned commit as a formalization and counts it as no formalized evidence, and the statement file is not a formalization link.

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Linked library material

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Linked from (16)

problems/integer_sequences/E0438/claims Problem 587 Nguyen and Vu's bound on square-sum-free sets Sequences and Densities of Integers Problem 439 Sequences and Densities of Integers integer_sequences/erdos_1977_differences_sums_integers_ii Remark (p. 209): the squares and the shifted primes are not sum intersector sets, and the 1/3 guess integer_sequences/khalfalah_2002_tight_bound_density_sum_no_two_perfect_square On the density of sequences of integers the sum of no two of which is a square. II. General sequences Theorem B (p. 2): a subset of [1,N] no two distinct elements of which sum to a square has at most .475N elements for N >= N_0 Theorem C (p. 7): circle-method count of representations of 2n by the alternating diagonal form in 2s+1 distinct variables from [(1-eps)M, M]number_theory/erdos_1980_old_new_problems_results_combinatorial_number_theory number_theory/erdos_1980_survey_problems_combinatorial_number_theory ramsey_theory/erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk Bound (p. 156, unnumbered): h(n) >= n/3, the largest subset of [1, n] with no two distinct members summing to a square

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